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Hey Nelly,
I love this question - it's such a classic GMAT exponent problem:
(1/5)^m (1/4)^18 = 1/ 2(10)^35 What is m?
I always look at exponent problems with three guiding principles:
1) In order to apply almost any rule related to exponents, you need to have like bases, so you should prime-factor each base to make them more maneuverable (we'll use this in a second)
2) All exponent rules relate to multiplication/division, so if a problem involved exponents and addition/subtraction, you should factor common terms to "manufacture" some multiplication.
3) Exponents are, by nature, "repetitive multiplication", so they lend themselves extremely well to patterns. So you can often establish patterns with small numbers and then extrapolate that pattern to the given numbers.
Here we'll use the first guiding principle to turn 1/4 and 1/10 into prime bases; 4 = 2^2 and 10 = 2*5, so we can express 1/4 and 1/10 in prime form to get:
(1/5)^m (1/4)^18 = 1/ 2(10)^35
(1/5)^m (1/(2^2))^18 = 1/[2(2*5)^35]
Now our bases are all in terms of 5 and 2, so we can much better combine those. Removing the parentheses, we'll distribute the exponents 18 and 35 to each term within them:
(1/5)^m (1/2)^36 = 1/2 (2^35 * 5^35)
Now, note that on the right hand side of the equation that 2 in the denominator has two 2 terms: 2, and 2^35. We can combine those to get 2^36:
(1/5)^m (1/2)^36 = 1/(2^36 * 5^35)
The 1/2^36 terms on both sides of the equation cancel each other out, leaving:
1/5^m = 1/5^35
So m must equal 35.
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